Robe i, Paolo
Wo king Pape
Lobbying in a mul idimensional policy space wi h salien
issues
Quade ni - Wo king Pape DSE, No. 922
P o ided in Coope a ion wi h:
Uni e si y o Bologna, Depa men o Economics
Sugges ed Ci a ion: Robe i, Paolo (2014) : Lobbying in a mul idimensional policy space wi h salien
issues, Quade ni - Wo king Pape DSE, No. 922, Alma Ma e S udio um - Uni e si à di Bologna,
Dipa imen o di Scienze Economiche (DSE), Bologna,
h ps://doi.o g/10.6092/unibo/amsac a/3946
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/159761
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ISSN 2282-6483
Lobbying in a mul idimensional
policy space wi h salien issues
Paolo Robe i
Quade ni - Wo king Pape DSE N°922
Lobbying in a mul idimensional policy space
wi h salien issues
Paolo Robe i∗
Janua y 27, 2014
Abs ac
We p esen a ci izen-candida e model on a mul idimensional pol-
icy space wi h lobbying, whe e ci izens ega d some issues mo e salien
han o he s. We ind ha special in e es g oups ha lobby on less
salien opics mo e he implemen ed policy close o hei p e e ed
policy, compa ed o he ones ha lobby on mo e salien issues. When
we in oduce wo ypes o ci izens, who di e wi h espec o he
salience o issues, we ind pooling equilib ia whe e o e s a e no able
o o se he e ec o lobbying on he implemen ed policy. This esul
is in sha p con as wi h p e ious wo k on unidimensional ci izen-
candida e models ha p edic he i ele ance o lobbying on he im-
plemen ed policy. In an ex ension o he model we p o ide ci izens
wi h he possibili y o gi ing mone a y con ibu ions o lobbies in o -
de o inc ease hei powe . Wi h mo e han one lobby pe dimension
we ha e wo indings. Fi s , unde some condi ions only he mos
ex eme lobbies ecei e con ibu ions. Second, he e ec i eness o a
lobby is maximized when he salience o an issue is low in he popu-
la ion and high o a small g oup o ci izens.
JEL-Classica ion: D72, D74, D78
Keywo ds: o ing, lobbying, salience, ci izen-candida e
1 In oduc ion
In 2012 in he US 3.30 billion dolla s we e spen on lobbying he Cong ess
and ede al agencies. In 2012 he e we e 12411 unique, egis e ed lobbying
∗pos doc o al esea ch ellow, depa men o economics. Uni e si y o Bologna. Piazza
Sca a illi 2, 40126 I aly, I aly. E-mail: paolo. obe [email protected]
1
i ms in he US1. The amoun o esou ces de o ed o his ac i i y and he
numbe o i ms in ol ed shows he ele ance o lobbying in he policy making
p ocess. In he poli ical economy li e a u e wi h a ixed numbe o candi-
da es, he equilib ium policy, esul ing om he in e ac ion o he o ing and
lobbying p ocesses, is de e mined by he maximiza ion o a weigh ed sum o
he u ili y unc ion o lobbies and some agg ega e wel a e unc ion o o e s,
see G ossman and Helpman (1996). Lobbying he e o e, in hese models, has
always an e ec on he implemen ed policy.
A ecen li e a u e, ini ia ed by Besley and Coa e (1997) and Osbo ne and
Sli inski (1996), has endogenized he numbe o candida es, allowing poli i-
cians o be selec ed, by majo i y o ing, among hose ci izens who choose o
en e he elec o al campaign. The ci izen-candida e amewo k was mean o
p o ide use ul insigh s on he endogenous posi ions o candida es and hei
numbe . None heless he ci izen-candida e model wi h lobbying, in oduced
by Besley and Coa e (2001) on a unidimensional policy space, p edic s ha
lobbies do no ha e an e ec on he equilib ium policy. Indeed o e s can
always suppo candida es wi h o se ing policy p e e ences, hus lobbying
changes he iden i y o he elec ed poli ician, bu no he implemen ed policy.
Conside ing ha he possibili y o inding o se ing candida es is inhe en
o ci izen-candida e models, i seems ha hey a e no i o unde s anding
lobbying.
In his pape we o e come his limi a ion, in es iga ing a ci izen-candida e
model wi h lobbying on a mul idimensional policy space. A mul idimensional
policy space is a e y ealis ic en i onmen o s udying he in e ac ion be-
ween o e s and candida es, because ci izens uly ha e p e e ences on many
di e en issues, om axa ion o en i onmen al opics and mo al alues. All
hese ma e s a e subjec o he ac ion o elec ed poli icians.
Mul idimensionali y inno a es Besley and Coa e (2001) because, wi h
many opics in he policy space, i is na u al o di e en ia e hem based
on hei salience. Indeed in e e y na ional and local poli ical ace o e s
conside some issues mo e impo an han o he s. Fo example in he Uni ed
S a es in he 2012 he s a e o he economy was impo an o 92% o e s2.
Issues like gay ma iage and abo ion was ins ead impo an o 38 % o e s.
In he model p esen ed he e we in oduce wo ypes o ci izens, di e en ia ed
by hei anking o issues. Fo example, one ype gi es mo e impo ance o
s a e o he economy, while he o he ype conside s he mo al issue he mos
ele an . The ype is p i a e in o ma ion o he ci izen. S ill o each ype
ci izens ha e he e ogenous p e e ences o policies in each dimension, e.g. o
1h p://www.opensec e s.o g/lobby/index.php
2h p://www.gallup.com/poll/153029/economy-pa amoun -issue- o e s.aspx
2
he ype ha conside s he s a e o he economy he mos ele an , he e a e
ci izens who belie e in s a e in e en ion, and o he s who hink he e should
be mo e ma ke and less s a e. Gi en ha candida es a e ci izens, hey
also ha e ypes. To keep he analysis mo e in ui i e we in oduce a single
unidimensional lobby pe issue. When aced wi h con ibu ions om lobbies
a e elec ions, an elec ed poli ician o a ype ha conside s he economy
mo e ele an , will please mo e he lobby on he mo al issue, because he
policy p e e ences o he poli ician a e weake on ha opic. The e o e,
di e en ypes o poli icians implemen di e en policies. Going back o he
o ing s age, we p o e ha he e a e pooling equilib ia, in which ci izens
a e no able o iden i y he ype o he candida es, and o e on expec ed
policies. Hence hey will o se lobbying ei he oo much o oo li le, and as
an icipa ed, in equilib ium lobbying will ha e an e ec on he implemen ed
policy.
The e a e o he esul s ha low na u ally om he se ing o he game.
One o hem is ha i , o all ypes o ci izens, he salience o an issue is
lowe ed, hen he lobby ha wo ks on ha opic inc eases i s in luence on he
implemen ed policy. Thus he mos e ec i e lobbies a e he ones ha wo k
on he opics ha people ca e less abou . This esul p o ides an explana ion
o why poli icians a e mo e sensi i e o lobbying, and he e o e less sensi i e
o o e s’p e e ences, on some issues. Fo example, in Janua y 2003, 63 % o
Ame icans we e agains he US go e nmen ’s decision o in ading I aq3. In
2005, 64% o I alian o e s, 55% among igh wing ones, we e in a ou o ci il
unions, bu he pa liamen ejec ed he law p oposal4. S ill in I aly, in 2010,
he pa liamen o ed laws o building new nuclea plan s and p i a izing he
public wa e sys em. Ne e heless, a ci izen ini ia i e in 2011 b ough 54 %
o he i alian o ing popula ion o o e on hese issues, and 96% o ci izens
who showed up o ed o he ejec ion o hese laws.
Ano he inding ha eme ges om he equilib ium analysis is ha some
ci izens, wi h he same mos p e e ed policy bu wi h a di e en anking
o issues, o e o di e en candida es. An example e e ed o he Ame i-
can P esiden ial elec ions o 2012 would be wo ci izens, bo h wishing mo e
income edis ibu ion and agains legal abo ion, who o ed o di e en can-
dida es, because one hough he economy was mo e impo an han mo al
issues and suppo ed Obama, while he o he ci izen el he opposi e and
o ed o Romney.
In ou game we ind also sepa a ing equilib ia, in which only one ype pe
candida e has an incen i e o en e he elec o al campaign. These equilib ia
3h p://www.cbsnews.com/s o ies/2003/01/23/opinion/polls/main537739.sh ml
4h p://www. epubblica.i /2005/i/sezioni/poli ica/p odipacs/i a a o/i a a o.h ml? e =sea ch
3

open he way o 3 candida es’ equilib ia, which we e excluded in Besley
and Coa e (1997). Indeed in he seminal pape o Besley and Coa e (1997)
s a egic o e s ne e ga e hei o e o a hi d candida e, e en hough he
was he closes o hem, because hey p e e ed o elec o su e hei second
mos p e e ed candida e. In ou game he e can be a hi d candida e ha
in equilib ium en e s because he hopes ha a leas one o he wo o he
unning candida es will wi hd aw, making him win o su e. An example
would be Ma io Mon i, who en e ed he I alian na ional elec o al campaign
in 2013, in he hope o no ha ing Be lusconi as a compe ing candida e. In
his case Mon i would ha e had a high p obabili y o winning he elec ions,
because he e would no be any candida e on he igh . Bu i Be lusconi
was o un, which he did, Mon i would ha e los .
An ex ension o he model pa ially endogenizes he powe o lobbies.
We p o ide ci izens wi h he possibili y o gi ing mone a y con ibu ions o
lobbies in o de o inc ease hei abili y o mo e he implemen ed policy
close o hei bliss poin . Wi h mo e han one lobby pe issue we ind ha ,
unde some condi ions, only he mo e ex eme lobbies in e e y dimension
ecei e con ibu ions. When s udying he e ec o he salience o an issue on
ci izens’ con ibu ions o lobbying, we ind ha he e ec i eness o a lobby
is maximized when he salience o a opic is low o mos o ci izens and high
o a small g oup. This small g oup is indeed he special in e es g oup he
inances he lobby.
All he esul s men ioned abo e a e de i ed om he h ee main ing e-
dien s o he model. The main con ibu ion o his pape is hus o b ing
oge he in a ci izen candida e model lobbying, mul idimensionali y o he
policy space and salience o issues.
The pape is o ganized as ollows: sec ion 2makes a li e a u e e iew on
o ing and lobbying. Sec ion 3in oduces he model. Sec ion 4p esen s he
esul s, sec ion 5endogenizes lobbying, while sec ion 6concludes.
2 Li e a u e Re iew
An ex ended li e a u e exis s on o ing and lobbying, in mos o i lobby-
ing is modeled h ough menu auc ions: he poli ician ecei es con ibu ions
con ingen on he implemen ed policy. See Be nheim and Whins on (1986),
G ossman and Helpman (1996) and Besley and Coa e (2001). The ci izen
candida e model has been de eloped sepa a ely by Osbo ne and Sli inski
(1996) and Besley and Coa e (1997). Osbo ne and Sli inski (1996) s udy he
model on a single dimension and assumes since e o ing, while Besley and
Coa e (1997) p o e hei esul s on a mul idimensional se ing wi h s a egic
4
o e s. Besley and Coa e (2001) ake he one dimensional ci izen candida e
model and add lobbies. In his pape in equilib ium lobbying is always o se
by he o e s, who o esee he subsequen lobbying and s a egically delega e
undoing he wo k o lobbies. E en hough lobbies pay hei con ibu ions and
he e is an e ec o in e es g oups on he choice o candida es, he e is no
e ec on he implemen ed policy. We will s udy he ci izen candida e model
on a mul idimensional policy space, wi h salien issues, wi h s a egic o -
ing and lobbies. Mo eo e cohe en ly wi h he idea ha in o ma ion abou
he gene al salience o issues is incomple e, du ing he elec o al campaign
we assume ha he e a e di e en ypes o o e s, each o hem iden i ied
wi h a di e en anking o issues. I is no known which ype is a o e
by he o he ci izens. In his way we also add ess he s a egic delega ion,
showing ha in some cases he e is a isible o e ec o lobbying on he
implemen ed policy. Felli and Me lo (2006) s udy he in e ac ion be ween
o ing and lobbying on a unidimensional se ing whe e he elec ed poli ician
can choose which lobbies o ecei e con ibu ions om. They show an e -
ec o lobbying on policies. Glaese e al. (2005) a gue ha Republicans
and Democ a s ha e become inc easingly ex emis on he eligious issue,
o induce hei co e cons i uencies o show up and o e, and ha is caused
by a g owing eligious sen imen in he US. Reading his ac h ough he
lenses o ou model we should see an e ec o lobbying on non mo al ela ed
issues in hese las yea s. I is indeed ue ha o example he Bu e ule
was suppo ed by he 72 % o Ame icans, 53 % among Republicans 5. S ill
i was no app o ed by he Cong ess. K asa and Polbo n (2010) c ea ed
a mul idimensional bina y model wi h salien issues. They a gue ha pol-
icy spaces, o med by ini e and especially bina y choices on each issue, a e
common in elec o al campaigns and deli e mo e ealis ic esul s. In hei
se ing candida es s a wi h some ixed posi ions on some dimensions and
igh o swing o e s on o he s, whe e hey a e lexible. Besley and Coa e
(2008) analyze he posi i e ole o ci izens’ ini ia i es o e e enda, in o de
o b ing implemen ed policies close o he will o he majo i y. The easons
why, e en i a Condo ce winne policy in all dimensions exis s, i could no
be implemen ed, a e ha in all elec ions issues a e bundled oge he . They
iden i y h ee main channels: a di e gence be ween he eli e o a pa y and
he popula opinion on non salien issues, a g oup o o e s who o e as a
single issue o e on a mino i y iew, and when a mino i y iew is suppo ed
by an in e es g oup on a non salien issue. The hi d case is he one we
ocus on in his pape . Two pape s ha conside he salience o issues a e
5h p://poli ical icke .blogs.cnn.com/2012/04/16/cnn-poll-7-ou -o -10-suppo -
bu e - ule/
5
Roeme (1998) and Lee and Roeme (2006). Roeme (1998) in es iga es wi h
a heo e ical model why a edis ibu i e poli ical pa y could be o ced o
p opose a low ax a e, in a wo ld wi h wo issues ( ax policy and eligion), i
he eligious dimension becomes e y salien . Lee and Roeme (2006) s udy
empi ically how acism among o e s con ibu ed o educe he income ax
a e in he US in he pe iod 1976-1992. While he easoning behind Roeme
(1998) elies on a speci ic dis ibu ion o o e s in he policy space, in pa ic-
ula on he p esence o a la ge poo acis pa o he popula ion, he esul s
o ou model explain he same phenomenon analyzed by Roeme (1998) and
a e alid o any dis ibu ion o ci izens in he policy space.
Finally he e is an impo an pa o he poli ical economy li e a u e
dedica ed o he empi ical s udy o he dimensions o he poli ical space and
he bliss poin s o MPs. Poole and Rosen hal (1985) use da a on he oll call
o ing on he US House and Sena e o es he p og am NOMINATE on he
posi ions o MPs in a unidimensional model. Poole and Rosen hal (1997)
and Poole and Rosen hal (2001) in eg a e his ini ial wo k wi h a dynamic
p og am, es ing o he numbe o dimensions o he poli ical space o MPs.
They ind ha he mul idimensionali y o he policy space can be educed
o 2. Hix e al. (2006) es he same s a is ical model on he Eu opean
Pa liamen . The concep o policy space in ou model is di e en om he
poli ical space o Poole and Rosen hanl. They collapse he policy space in a
2-dimensional poli ical space because mos o he imes MPs o e ollowing
hei pa y di ec ion on he conse a i e-libe al axis, while on o he o es
hei beha io can be eg ouped ollowing he No h-Sou h US axis. The
numbe o dimensions is hus ela ed o he numbe o opposi e pa liamen a y
blocks ha appea in di e en o es. Ins ead he dimensions o he policy
space in ou model e e o di e en issues ha a ec o e s. We do no
model pa liamen a y o ing and assume a single poli ician implemen s a
mul idimensional policy when she is elec ed.
3 The Model
The K−dimensional policy space is deno ed by D= [0,1]K. The se o
ci izens is deno ed by N={1, ..., M}, wi h Me en. Ci izen i∈Nhas he
ollowing u ili y unc ion:
U(q, i) =
K
X
k=1
λi
ku(qk, qi
k) + ρyi,(1)
6
whe e qkis he k h elemen o he ec o policy q∈D.u(qk, qi
k) is s ic ly
conca e in qk, single-peaked and symme ic a ound qi
k.qiis ci izen i’s bliss
poin . ρmeasu es he in ensi y o ci izen’s p e e ences o e money wi h
espec o policy. λi
k≥0 is he weigh gi en by ci izen i o dimension o
issue k. Fo e e y M, Mis he s ep densi y unc ion ha desc ibes he
dis ibu ion o he M o e s’ bliss poin s in he policy space. FMis he
cumula i e induced measu e unc ion o M. We assume ha FMcon e ges
in dis ibu ion o a cumula i e unc ion F. The Radon-Nikodym de i a i e
o Fis , i.e. F(A) = RA dµ, A ⊂D, and µis he Lebesgue meausu e in
RK. We also assume ha > 0 almos e e ywhe e, ha means ha he
popula ion o ci izens is “dense” in he policy space.
Lobbies ha e a simila u ili y unc ion. Fo simplici y we assume ha
he e is jus one lobby o each dimension:
V(q, k) = µku(qk, qL
k) + yL
k,(2)
whe e µkis an idiosync a ic pa ame e o e e y lobby. µkis he ela i e
in ensi y o lobby’s p e e ences o policy wi h espec o money. Fo he
same u ili y gain a lobby wi h highe µkis willing o pay mo e. We no malize
ans e s o be ze o RSyidi +PK
k=1 yL
k= 0.
3.1 Unce ain y abou o e s’ p e e ences
Some pa s o ou analysis will be es ic ed o K= 2. In his se ing we will
assume ha he e a e only wo ypes o ci izens: ype 1 is cha ac e ized by
weigh s (1, λ1
2), ype 2 by (1, λ2
2), whe e λ1
2< λ2
2, meaning ha ype 1 ci izens
weigh mo e dimension 1 wi h espec o ype 2 ci izens. The ypes se is
deno ed by T={1,2}. Ci izens o ype 1 and 2 ha e he same dis ibu ion
on he policy space. The ype o each ci izen is no known a he beginning,
he e is a common p io : each ci izen has p obabili y po being o ype 1
and p obabili y 1 −po being o ype 2. We pa ame e ize λ1
2=θη and
λ2
2= (2 −θ)η,θ < 1, whe e η= 1/2(λ1
2+λ2
2) is he a i hme ic a e age
o he wo pa ame e s. Unce ain y abou ypes o o e s makes his game
Bayesian.
3.2 En y o candida es
Each ci izen can en e as a candida e paying a small cos c. We deno e by
σ(i, ) : S×T→ {0,1} he decision o ype ci izen i,σ(i, ) = 1 indica es
(i, )’s decision o en e as a candida e, while i σ(i, ) = 0 (i, ) will s ay ou .
We de ine C(σ) = {i∈S:∃ such ha σ(i, ) = 1} he se o candida es o
7
Figu e 3: Pa i ion o he policy space wi h 2 ypes o o e s
The do s Aand B ep esen candida es Aand B’s expec ed policies. The
hype planes h1λ1
2and h1λ2
2pa i ion he policy space in 4 a eas. As we can see
he e a e wo a eas, Wand Y, whe e he wo ypes o same ci izen o e o
he same candida e. The e a e o he wo a eas, Xand Z, whe e he wo ypes
o e o di e en candida es. In a ea Xe e y ci izen o es wi h p obabili y p
o Aand wi h p obabili y 1−p o B. In a ea Y he con e se is ue. Ci izens
in his a eas can be somehow called swing o e s, because he bene icia y o
hei o e is no known o o he ci izens and he candida es. Swing o e s a e
no necessa ily mode a e, so hei p e e ed policy could be no “in be ween”
he candida es’ expec ed policies. In a ea Xwe a e in e es ed o know he
p obabili y densi y unc ion o he p opo ion o ci izens o ing o A, which
is equi alen o compu e he densi y o he a iable Pn
i=1
Xi
n o n→ ∞,
whe e Xiis he Be noullian a iable aking alue 1 i ci izen iwi h bliss
poin qi∈Xis o ype 1. Fo he law o La ge Numbe s we know ha he
limi con e ges almos su ely o he expec ed alue E(X1) = E(X2) = ... =p.
We hen know ha he e en “o all ci izens wi h bliss poin in X p o hem
14

o e o A” has p obabili y 1. Candida e A ecei es he ollowing o es:
ZW
(x)dx +pZX
(x)dx + (1 −p)ZZ
(x)dx,
while candida e B ecei es:
ZY
(x)dx +pZZ
(x)dx + (1 −p)ZX
(x)dx.
Nex ollow lemmas ha desc ibe he subgame equilib ia in he o ing s age.
We will de ine a “winning candida e” a candida e who has a posi i e p oba-
bili y o winning o ying. When we will say ha o e s “ ace” a ce ain se
o candida es, we e e o he si ua ion o ha ing ha se o candida es in
he o ing s age. In he nex sec ion i will be clea ha he e a e equilib ia
o he game, in which he se o ci izens ha choose o en e as candida es is
la ge han he se o candida es ha ge o he o ing s age, because some
o hese candida es will no be d awn by Na u e.
Lemma 2 I he e is an in ini e numbe o o e s, and o e s ace wo can-
dida es Aand B, a necessa y and su icien condi ion such ha one o hem
does no lose wi h ce ain y is he ollowing:
ZW
(x)dx+pZX
(x)dx+(1−p)ZZ
(x)dx =ZY
(x)dx+pZZ
(x)dx+(1−p)ZX
(x)dx,
(10)
whe e (W, X, Y, Z) esul om he pa i ion o he space by wo sepa a ing
hype planes, h1,λ1
2and h1,λ2
2. In Wci izens o bo h ypes o e o A, in Y
ci izens o bo h ypes o e o B, in X ype 1ci izens o e o Aand ype
2ci izens o e o Band he con e se in Y. I condi ion 10 is sa is ied
candida es ie.
I he candida es’ expec ed policies q∗Aand q∗Ba e such ha q∗A
i=q∗B
i
o ei he i∈ {1,2}h1,λ1
2and h1,λ2
2coincide and equa ion 10 becomes:
ZW
(x)dx =ZY
(x)dx. (11)
Lemma 3 Fo e e y  > 0 he e exis s M, such ha i he numbe o ci izens
Msa is ies M > M, i condi ion 10 is sa is ied and o e s ace candida e
Aand B, he p obabili y ha candida e Awins, PW
A, sa is ies he ollowing
inequali y:

PW
A−1
2
< . (12)
15
The p obabili y ha candida e Aloses, PL
A, sa is ies he ollowing inequali y:

PL
A−1
2
< . (13)
The p obabili y ha candida e A ies, PT
A, sa is ies he ollowing inequali y:
PL
A< . (14)
Unde he same condi ions bu condi ion 10, o one candida e Pamong
{A, B} he p obabili y o winning PW
Psa is ies he ollowing inequali y:
PW
P−1< . (15)
The p oo is p esen ed in he appendix. Lemma 2says ha , wi h an in ini e
numbe o o e s, a necessa y condi ion o a candida e no o lose o su e in
he o ing subgame is ha he wo candida es spli in hal he cos i uency.
Hence he candida es ie. Lemma 3says ha , o a ini e bu high M,
unde condi ion 10 he p obabili y o winning and losing app oxima es one
hal , ha gi es o each candida e he same payo o ying. I condi ion
10 is no sa is ed one candida e loses wi h a p obabili y ha app oxima es
one. The e o e wi h a high bu ini e numbe o o e s M, only he subgame
equilib ia ha exis o an in ini e numbe o o e s su i e. We will he e o e
es ic ou analysis o equilib ia o he ones wi h an in ini e numbe o o e s.
The nex lemmas s a e condi ions abou he en y o a hi d candida e.
Lemma 4 The e a e no subgame o ing equilib ia, whe e o e s ace 3can-
dida es such ha all h ee candida es ha e a posi i e p obabili y o winning
o ying.
The p oo o lemma 4is gi en in Besley and Coa e (1997). Indeed i he
popula ion o o e s is dense in he policy space, ha is ou case, a subgame
equilib ium whe e h ee candida es ha e a posi i e p obabili y o winning
canno exis because a o e ha is nea ly indi e en be ween wo candida es
will a he de ia e and o e o he second p e e ed candida e o make him
win wi h p obabili y one.
Lemma 5 The e is a o ing subgame whe e ci izens ace 2candida es whose
expec ed implemen ed policies sa is y condi ion 10, and ano he candida e o
whom i does no , such ha he hi d candida e loses wi h ce ain y.
The p oo is p esen ed in he appendix.
While lemma 4s a es ha he e canno be subgame equilib ia wi h 3 winning
candida es, lemma 5says ha subgame equilib ia wi h 2 winning candida es
exis .
16
4.3 En y Equilib ium
To cha ac e ize a wo-candida e equilib ium we need o s udy he Bayesian
na u e o he en y s age. We know ha a necessa y condi ion o a wo
ying candida e equilib ium is gi en by equa ion 10. In equa ion 10 he ou
a eas a e de ined by he hype planes, which a e based on he candida es’
expec ed policies. Le us de ine σ∗ he equilib ium en y unc ion. I o all
i∈C(σ∗)σ∗(i, 1) = σ∗(i, 2) he en y equilib ium is de ined o ally pooling.
I o all i∈C(σ∗)σ∗(i, 1) 6=σ∗(i, 2) he en y equilib ium is de ined o ally
sepa a ing. O he wise we call he en y equilib ium pooling. I he en y
equilib ium is o ally sepa a ing, he expec ed policies a e he implemen ed
policies, i he en y equilib ium is o ally pooling, he expec ed policies a e
he expec ed implemen ed policies. In his sec ion wi h abuse o no a ion
we e e o q∗i as he policy implemen ed by ype candida e ii she we e
elec ed, and o P i na u e has d awn ype o candida e P. We e e
also o he Euclidean dis ance in R2be ween xand yas |x−y|. We deno e
− as he non ype. When he e a e only wo candida es we deno e −P
as he non Pcandida e. The e is an in ini e amoun o en y equilib ia
gi en by he posi ions o he candida es. In a game wi hou unce ain y
abou he ypes o candida es, and wi hou unce ain y abou he o ing
beha io , a necessa y condi ion o a 2 candida es equilib ium is ha he 2
candida es ie. O he wise one candida e would lose o su e and would a he
no en e and sa e c. Mo eo e when he popula ion o s a egic o e s is
dense in he policy space Besley and Coa e (1997) p o e ha he e canno be
equilib ia wi h mo e han 2 candida es. In ou game he unce ain y abou
o ing beha io is sol ed wi h he law o la ge numbe . Fo wha conce ns
unce ain y abou he ypes o candida es, i we ha e a pooling equilib ium
we know ha in he o ing s age ci izens will ace 2 candida es, e en hough
hey do no know hei iden i y. In his case he expec ed implemen ed
policies o he candida es mus spli he elec o a e e enly. Ins ead, i we
ha e a sepa a ing equilib ium, i could be ha he ype ha is supposed o
en e o a ce ain candida e is no d awn by Na u e. The e o e ci izens in
he o ing s age would ace only one candida e. This changes he incen i es
o en e as a candida e, because a candida e ha would lose agains i s
opponen could ind con enien o en e , indeed wi h some p obabili y he
o he candida e is no d awn and she wins wi h ce ain y. Consequen ly
he e will be sepa a ing equilib ia wi h 2 ying candida es, and wi h 2 non
ying candida es. The same easoning opens he way o sepa a ing equilib ia
wi h mo e han 2 candida es.
In he nex heo ems we s a e he condi ions such ha 2 candida es ind
con enien o en e . We de ine he ollowing quan i y:
17
d(q, ) := p(q1− 1)2+ (q2− 2)2, whe e , q a e 2-dimensional ec o s. dis
he euclidean dis ance.
Condi ion 1 (s ong non p oximi y) A wo-candida e equilib ium, C(σ∗) =
{A, B}, sa is ies s ong non p oximi y i he ollowing condi ions a e sa is ied:
1
2U(q∗A , A )−U(¯qB, A ) + p(1 −p)d2(q∗Bs, q∗B−s)> c, (16)
1
2U(q∗Bs, Bs)−U(¯qA, Bs) + p(1 −p)d2(q∗Bs, q∗B−s)> c,
1
2U(q∗A− , A− )−U(¯qB, A− ) + p(1 −p)d2(q∗A , q∗A− )> c,
1
2U(q∗B−s, B−s)−U(¯qA, B−s) + p(1 −p)d2(q∗A , q∗A− )> c,
whe e U(q∗i, i)includes he lobbies’ con ibu ion.
Condi ion 2 (non p oximi y) A wo-candida e equilib ium, C(σ∗) = {A , Bs},
sa is ies non p oximi y i he ollowing condi ions a e sa is ied:
1−ps
2U(q∗A , A )−ps
2U(q∗Bs, A )> c + (1 −ps)U(qsq, A ),(17)
1−p
2U(q∗Bs, Bs)−ps
2U(q∗A , Bs)> c + (1 −p )U(qsq, Bs),
1−ps
2U(q∗A− , A− )−ps
2U(q∗Bs, A− )< c + (1 −ps)U(qsq, A− ),
1−p
2U(q∗B−s, B−s)−ps
2U(q∗A , B−s)< c + (1 −p )U(qsq, B−s),
and o all ci izens ∈N, ha would win wi h ce ain y pai wise agains
ei he A o Bs, he ollowing condi ion is sa is ied:
(1 −p ps)U(q∗ , )< c+p (1−ps)U(q∗A , )+ps(1−p )U(q∗Bs, )+(1−ps)(1−p )U(qsq, ),
(18)
o all ci izens ∈N, ha would lose wi h ce ain y pai wise agains bo h A
o Bs, he ollowing condi ion is sa is ied:
(1 −ps)(1 −p )U(q∗ , )< c + (1 −ps)(1 −p )U(qsq, ),(19)
o all ci izens ∈N, ha would lose wi h ce ain y agains A and win
agains Bs, he ollowing condi ion is sa is ied:
(1 −p )U(q∗ , )< c + (1 −p )psU(q∗Bs, ) + (1 −ps)(1 −p )U(qsq, ),(20)
o all ci izens ∈N, ha would win wi h ce ain y agains A and lose
agains Bs, he ollowing condi ion is sa is ied:
(1 −ps)U(q∗ , )< c + (1 −ps)p U(q∗A , ) + (1 −ps)(1 −p )U(qsq, ),(21)
18
whe e p and psa e he p io p obabili ies espec i ely o ypes and s, and
U(q∗i, i)includes he lobbies’ con ibu ion.
Theo em 1 ( o ally pooling) A wo-candida e equilib ium, C(σ∗) = {A, B},
exis s and is o ally pooling i and only i condi ion 1is sa is ied and he wo
expec ed implemen ed policies ¯qA=pq∗A1+(1−p)q∗A2and ¯qB=pq∗B1+(1−
p)q∗B2gene a e hype planes h1,λ1
2and h1,λ2
2 ha sa is y equa ion 10.
Theo em 1says ha a su icien condi ion o a wo-candida e o ally
pooling equilib ium is ha he expec ed implemen ed policies spli in hal
he elec o a e. All ypes o candida es Aand B ind p o i able o en e
because hey ha e 1/2 p obabili y o win and o condi ion 1 hey a e be e
o han le ing he o he candida e win.
Theo em 2 ( o ally sepa a ing) A wo ying candida e equilib ium, C(σ∗) =
{A , Bs}, exis s and is o ally sepa a ing i condi ion 2is sa is ied and he wo
policies q∗A and q∗Bsgene a e hype planes h1,λ1
2and h1,λ2
2 ha sa is y equa-
ion 10.
Theo em 2says ha a su icien condi ion o a wo-candida e o ally sep-
a a ing equilib ium is ha na u e selec s ypes whose implemen ed policies
spli in hal he cons i uency. Mo eo e o condi ion 2only one ype pe
candida e mus ind con enien o en e .
19

Figu e 4: Posi ions o a 2 candida es o ally pooling equilib ium
Wi h abuse o no a ion we de ine he implemen ed policy q∗(ρ) in equilib-
ium as depending on he pa ame e ρ, he p e e ence o money o ci izens,
while keeping all o he pa ame e s cons an .
P oposi ion 4 In o ally pooling equilib ia he in e es g oups’ lobbying has
an e ec on he implemen ed policy, ha is
q∗(ρ)6=q∗(0),
o e e y ρ > 0.
The en y equilib ium analysis deli e s se e al esul s: i s o all, di -
e en ly om Besley and Coa e (1997) we ha e an e ec o lobbying on im-
plemen ed policies i he equilib ium is o ally pooling. Indeed in a o ally
pooling en y equilib ium bo h ypes o he same candida e en e , o e s do
no know which ype hey ace so hey o e on expec ed policies. Depending
on he ype o candida e ealized, hey will ha e o se ei he oo much o oo
li le. The e o e lobbying can ma e o implemen ed policies, and in ou
model he channel is he incomple e in o ma ion abou gene al salien issues
in he elec o al campaign. The di e ence be ween a o ally pooling and a
o ally sepa a ing equilib ium is ha in he la e he candida es en e ing
20
a e signalling hei ype, no only wi h hei ac ion o en e ing, bu also wi h
he opponen ’s ac ion.
4.4 3candida es equilib ium
He e we speci y he condi ions o a 3 candida es equilib ium. The e we e no
3 candida es equilib ia in Besley and Coa e (1997). A necessa y condi ion
o ha e 3 candida es equilib ia in he se ing o Besley and Coa e (1997) is
ha all candida es ie in he o ing s age. Bu wi h s a egic o ing and
wi h a high numbe o ci izens ”dense” in he policy space, all o e s ha
a e nea ly indi e en be ween hei mos liked candida e and he second one,
will de ia e and o e o he second p e e ed one o make his candida e win
wi h ce ain y.
In ou model ins ead he unce ain y abou ci izens’ salience ansmi s o
unce ain y abou o ing beha io , abou candida es’ implemen ed policies,
and, in a sepa a ing equilib ium, abou candida es’ en y. Indeed, in a sepa-
a ing equilib ium, he en y condi ions o a candida e a e sa is ied o only
one ype. I Na u e ex ac s he o he ype, i will no en e . The e o e his
unce ain y abou en y o candida es gi es oom o sepa a ing equilib ia
wi h mo e han 2 candida es, whe e some candida es en e hoping ha a
leas one o he wo ying candida es is no ex ac ed by Na u e.
The nex heo em s a es he necessa y and su icien condi ions o a 3 can-
dida es equilib ium, whe e he e a e wo candida es who ie in he o ing
s age, and a “ hi d candida e” who would lose wi h ce ain y i all 3 candi-
da es en e , bu wins pai wise agains bo h o he candida es.
Theo em 3 A3candida es equilib ium, wi h candida es (A , Bs, Cl)exis s
i A and Bsimplemen ed policies q∗A and q∗Bsgene a e hype planes ha
sa is y equa ion 10, and he ollowing condi ions a e sa is ied:
hpj
2+ (1 −pj)(1 −pl)iU(q∗Pi, Pi) + hpj
2−(1 −pl)pjiU(q∗−Pj, Pi)>
c+pjplU(q∗Cl, Pi) + (1 −pl)(1 −pj)U(qsq, Pi),(22)
hpj
2+ (1 −pj)(1 −pl)iU(q∗P−i, P−i) + hpj
2−(1 −pl)pjiU(q∗−Pj, P−i)<
c+pjplU(q∗Cl, P−i) + (1 −pl)(1 −pj)U(qsq, P−i),(23)
o i, j ∈ { , s}and Pi∈ {A , Bs}, whe e pjis he p obabili y o ype j.
The ollowing condi ions mus be sa is ied o a candida e Cl, who would win
21
wi h ce ain y pai wise agains bo h A and Bs:
(1 −p ps)U(q∗Cl, Cl)> c +p (1 −ps)U(q∗A , Cl) +
ps(1 −p )U(q∗Bs, Cl) + (1 −p )(1 −ps)U(qsq, Cl),(24)
(1 −p ps)U(q∗C−l, C−l)< c +p (1 −ps)U(q∗A , C−l) +
ps(1 −p )U(q∗Bs, C−l) + (1 −p )(1 −ps)U(qsq, C−l).(25)
The ollowing condi ion mus be sa is ied o a ci izen , who would win
pai wise agains bo h A and Bs, and would lose agains Cl:
(1 −p ps)(1 −pl)U(q∗ , )< c +p (1 −ps)(1 −pl)U(q∗A , ) +
ps(1 −p )(1 −pl)U(q∗Bs, ) + (1 −p )(1 −ps)(1 −pl)U(qsq, ).(26)
The ollowing condi ion mus be sa is ied o a ci izen , who would win
pai wise agains A and Bsand Cl:
(1 −p ps)U(q∗ , )< c +p (1 −ps)(1 −pl)U(q∗A , ) +
ps(1 −p )(1 −pl)U(q∗Bs, ) + plU(q∗Pl, )(1 −p )(1 −ps)(1 −pl)U(qsq, ).(27)
The ollowing condi ion mus be sa is ied o a ci izen , who would win
pai wise agains i, and would lose agains ei he jo Cl:
(1 −pj)(1 −pl)U(q∗ , )< c +
(1 −pj)(1 −pl)piU(q∗i, ) + (1 −pi)(1 −pj)(1 −pl)U(qsq, ),(28)
o i, j ∈ {A , Bs}.
The ollowing condi ion mus be sa is ied o a ci izen , who would win
pai wise agains iand Cl, and would lose agains j:
(1 −pj)U(q∗ , )< c + (1 −pj)(1 −pl)piU(q∗i, ) +
(1 −pj)plU(q∗Cl, ) + (1 −pi)(1 −pj)(1 −pl)U(qsq, ),(29)
o i, j ∈ {A , Bs}.
The ollowing condi ion mus be sa is ied o a ci izen , who would lose
pai wise agains A , Bsand win agains Cl:
(1 −p )(1 −ps)U(q∗ , )< c +
(1 −p )(1 −ps)plU(q∗Cl, ) + (1 −p )(1 −ps)(1 −pl)U(qsq, ).(30)
The ollowing condi ion mus be sa is ied o a ci izen , who would lose
pai wise agains A , Bsand Cl:
(1 −p )(1 −ps)(1 −pl)U(q∗ , )< c + (1 −p )(1 −ps)(1 −pl)U(qsq, ).(31)
22
5 Endogenizing lobbying
We p esen he e an ex ension o he model whe e ci izens can in e ac di ec ly
wi h lobbies, gi ing hem mone a y con ibu ions in o de o inc ease hei
powe and hus ob ain a mo e a o able implemen ed policy.
We assume ha p e e ence in ensi y o policy wi h espec o money and
he salience o issues a e idiosync a ic, i.e. ρiand λi
j o ci izen i. We assume
also ha he e can be mo e han one lobby o e e y poli ical dimension.
Con ibu ion o lobbies is implemen ed a e elec ions a e o e . To simpli y
he analysis we also assume ha a e elec ions and be o e con ibu ion akes
place he ype o each ci izen is e ealed. I a subse R⊂No ci izens
con ibu es o lobby khe ela i e in ensi y o policy wi h espec o money
becomes:
µL
k:= bL
k+√yk,(32)
whe e yk:= Pi∈Ryi
k,yi
kis he mone a y con ibu ion o ci izen i o lobby k,
and bL
kis a posi i e cons an 6. We also de ine y−i
k:= Pj∈R,j6=iyj
k.
I ci izen icon ibu es yi=PK
k=1 yi
k≥0 o lobbying he u ili y becomes:
U(q, i) =
K
X
k=1
λi
ku(qk, qi
k)−ρiyi,
Ci izens con ibu e a e elec ions a e o e and be o e lobbies o e hei con-
ibu ion schedules o he elec ed poli ician.
I a ci izen con ibu es yi
k o lobby k he in e es g oup inc eases i s p e e -
ences o he policy, his has a posi i e e ec on he con ibu ion schedule
o e ed o he poli ician and hus on q∗P,mo ing i close o he bliss poin
o he lobby.
Le us de ine
yM
k(ρ, qk, λk) := ρPλP
kλk(qk−q∗
k)(qL
k−qP
k)
ρ(λP
k+ρPµL
k)22
.
P oposi ion 5 In equilib ium only a subse Rk⊂Ndona es o lobby k.
Ci izen ibelongs o Rki and only i (qi
k−q∗
k)(qL
k−qP
k)≥0and (ρi, qi
k, λi
k)∈
6We assume ha ci izens’ mone a y con ibu ion a ec s µL
kand hus he willingness o
pay wwi h a dec easing ma gin, he same esul s would be ob ained i we assume ha he
con ibu ion inc eases linea ly wand he ci izen’s mone a y cos is con ex. The d awback
o his las and mo e na u al o mula ion is ha o in e nal cohe ence also he lobby’s
mone a y cos would need o be con ex, wwould hen be conca e in he lobby’s u ili y
and he implemen ed policy q∗would need o be ecompu ed.
23
ze o is 1/√M, see Felle (2008), page 184. The Cen al Limi Theo em en-
su es ha √MhVMx+VMZ
M−pRx −(1 −p)RZ iapp oaches a No mal dis-
ibu ion wi h mean 0 and 1. The e o e he dis ibu ion o VMx+VMZ
Mwill be
close o symme ic dis ibu ion as Mg ows la ge . The e o e o any  > 0
he e exis s a Msuch ha o all M > M|PW
A−1/2|<  and |PL
A−1/2|< ,
and PT
A< .
P oo o Lemma 5: We now p o ide equilib ium s a egies in he o -
ing subgame o 2 and 3 candida es such ha a hi d candida e ne e inds
con enien o en e . We do no speci y wha a e he belie s o o e s on
he candida es’ implemen ed policies, because he p oo is alid o e e y
a ay o belie s ha a e common o all o e s. I C(γ) = {A, B}all non
indi e en ci izens o e o hei a o i e candida e and indi e en ci izens
do no o e. This ec o o s a egies is a subgame Nash equilib ium and
does no include weakly domina ed s a egies. I C(γ) = {A, B, C}, whe e
Aand B’s expec ed implemen ed policies sa is y condi ion 10, and o e s
ace h ee candida es8, equilib ium s a egies a e buil as ollows: all o e s,
including C, ha a e non indi e en be ween Aand B o e o hei a o i e
candida e among {A, B}. Vo e s ha a e indi e en be ween Aand Bbu
s ic ly p e e C o ei he Ao B o e o C. Vo e s ha a e indi e en
be ween Aand Bbu p e e ei he Ao B o Cspli : hal o hem o e o
Aand hal o hem o e o B. This ec o o s a egies is a subgame Nash
equilib ium and does no include weakly domina ed s a egies. Vo e s ha
a e no indi e en be ween Aand Bdo no change hei o e because hey
would make he o he candida e win. Vo e s ha a e indi e en be ween A
and Band s ic ly p e e Cge he same u ili y in equilib ium o ing o
A, B, C o no o ing. Bu o ing o Cis he only non weakly domina ed
s a egy hey ha e. Indeed o ing o Cmakes his kind o ci izen s ic ly
be e o han o ing o A, B o no o ing, when he e a e enough ci izens
who o e o Csuch ha she is pi o al. Ci izens ha a e indi e en be ween
Aand Band p e e ei he A, B o Cge he same u ili y in equilib ium
o ing o A, B, C o no o ing. Le us assume hey o e o A. Vo ing o
Ais no weakly domina ed by o ing o B, bu weakly domina es no o ing
and o ing o C. I he e a e enough ci izens who o e o Csuch ha his
kind o ci izen is pi o al she p e e s o ing o A(o B) han o ing o Co
no o ing. The o ing equilib ium s a egies in he 3 candida es’ subgame
a e buil such ha i Aand Bwe e ying when Cwas no unning, hey
a e s ill ying wi h C unning, because ci izens ha a e indi e en be ween
8This speci ica ion is needed, because no all candida es could be d awn by Na u e.
30

A, B ei he o e o Co hey spli equally among Aand B. Gi en hese
o ing equilib ium s a egies, when o e s ace all 3 candida es, Closes wi h
ce ain y.
P oo o Theo em 1: he o ing equilib ium is gua an eed by lemma
2, whe e expec ed policies a e ¯qA,¯qB. Now we check i any ype o A
and Bhas an incen i e o de ia e no en e ing as a candida e. Condi-
ion 1con ols o ha , indeed ype o candida e Pdoes no de ia e i
1
2U(q∗P , P ) + U(¯q−P, P )−c > U(¯q−P, P ).
P oo o Theo em 2: he o ing equilib ium is gua an eed by lemma 2,
whe e policies a e qA , qBs. Type o candida e P uns because ps
2[U(q∗P , P )+
U(q−Ps, P )]+(1−ps)U(q∗P , P )−c>psU(q−Ps, P )+(1−ps)U(qsq, P ). Type
− o candida e Pdoes no un because ps
2[U(q∗P− , P− ) + U(q−Ps, P− )] +
(1 −ps)U(q∗P− , P− )−c<psU(q−Ps, P− ) + (1 −ps)U(qsq, P− ). A hi d can-
dida e, ha wins agains ei he A o Bsdoes no ind con enien o en e
because psp
2[U(q∗A , ) + U(q∗Bs, )] + (1 −p ps)U(q∗ , )<psp
2[U(q∗A , ) +
U(q∗Bs, )] + c+p (1 −ps)U(q∗A , ) + ps(1 −p )U(q∗Bs, ) + (1 −ps)(1 −
p )U(qsq, ). By 4when acing bo h candida es loses o su e. The same
easoning applies o de ia ions om hi d candida es who a e winning jus
agains one candida e be ween A and Bs, o su e lose s.
P oo o P oposi ion 5The o al con ibu ion yk o lobby k ha max-
imizes ci izen i’s u ili y is:
yk=yM
k(ρi, qi
k, λi
k) = ρPλP
kλi
k(qi
k−q∗
k)(qL
k−qP
k)
ρi(λP
k+ρPµL
k)22
,(35)
whe e condi ion 35 is de i ed om he FOC o ci izen i’s u ili y. I he
sum y−i
ko o he ci izens’ con ibu ions is al eady la ge han he op imal
yM
k(ρi, qi
k, λi
k), ci izen idoes no con ibu e. The equilib ium con ibu ion o
lobby kis maxρ,qk,λkyM
k, ha ep esen s he op imal con ibu ion o ci izens
whose idiosync a ic pa ame e s ρ, qk, λka e he a g max o yM
k. All o he
ci izens do no con ibu e, because hei op imal o al con ibu ion is lowe
han maxρ,qk,λkyM
k.
31
